Towards a theory of gravitational radiation or What is a ...potor/nurowski_waves.pdf ·...

Recent LIGO announcementGravitational radiation theory: summary

Prehistory: 1916-1956History: 1957-1962

Towards a theory of gravitational radiationor

What is a gravitational wave?

Paweł Nurowski

Center for Theoretical PhysicsPolish Academy of Sciences

King’s College London,28 April 2016

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Plan

1 Recent LIGO announcement

2 Gravitational radiation theory: summary

3 Prehistory: 1916-1956

4 History: 1957-1962

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LIGO detection: Its relevance

the first detection ofgravitational waves

the first detection of ablack hole; of a binaryblack-hole; of a mergingprocess of black holescreating a new one; Kerrblack holes exist; blackholes with up to 60 Solarmasses exist;

the most energetic processever observed

important test of Einstein’s General Theory of Relativitynew window: a birth of gravitational wave astronomy

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The main ideas

Linearize Einstein’s field equations and look for waves in thelinearized theory (A. Einstein 1916, 1918)

Define what a plane wave is in the full theory (N. Rosen, A.Einstein 1937, I Robinson 1956(?), H Bondi 1957, H Bondi,F Pirani, I Robinson 1959)Define what a radiative spacetime is in the full theory:

by algebraical speciality of the Weyl tensor (F Pirani 1957)by the boundary conditions satisfied by the solutions of thevacuum Einsten’s equations (A Trautman 1958, H Bondi 1962,R Penrose 1962)by an existence of a special null vector field (I Robinson1961, I Robinson, A Trautman 1960, P Kerr 1962)by the requirement that waves carry information (ATrautman 1962)

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The main ideas




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The main ideas




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The main ideas




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The main ideas




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The main ideas




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The main ideas




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People: Albert Einstein (14.3.1879-18.04.1955)

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People: Nathan Rosen (22.3.1909-18.12.1995)

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People: Hermann Bondi (1.11.1919-10.9.2005)

H. Bondi with P. Bergman(Warsaw 1962)

H. Bondi with L. Infeld(Warsaw 1962)

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People: Ivor Robinson

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People: Ivor Robinson

H. Bondi with I. Robinson(Warsaw 1962)

I. Robinson with A. Trautman(Trieste 1985)

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People: Felix Pirani (2.2.1928-31.12.2015)

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People: Andrzej Trautman

A. Trautman with S.Chandrasekhar (Warsaw 1973)

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People: Roger Penrose

R Penrose (second from the right) with E T Newman (in thecenter); J. A. Wheeler on the left and C. Møller on the right(Warsaw 1973)12/48



Gravitational waves: Einstein 1916

Einstein, Albert,Naherungsweise Integration derFeldgleichungen derGravitation, 22.6.1916

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Einstein linearized hisfield equations Gµν = κTµν for the metric gµν assuming that

gµν = ηµν + γµν , γ′µν = γµν −12ηµνtrace(γαβ) ,

i.e. that the metric gµν is a slightly perturbed Minkowskimetric ηµν , with the relevant part of the perturbation given byγ′µν ,

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Einstein linearized hisfield equations Gµν = κTµν for the metric gµν assuming that

gµν = ηµν + γµν , γ′µν = γµν −12ηµνtrace(γαβ) ,

i.e. that the metric gµν is a slightly perturbed Minkowskimetric ηµν , with the relevant part of the perturbation given byγ′µν ,

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Gravitational waves: Einstein 1916 - the wave equation

he obtained

�γ′µν = 2κTµν

this, outside the sources, is the relativistic wave equation

�γ′µν = 0

for the perturbation, which justifies the claim that in thelinearized Einstein’s theory gravitational waves -perturbations of the spacetime metric traveling with speed oflight - do exist.

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he obtained



�γ′µν = 0


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he obtained



�γ′µν = 0


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he obtained



�γ′µν = 0


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he obtained



�γ′µν = 0


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Gravitational waves: Einstein 1916 - quadrupole formula

Einstein has also shownthat in the linearized theoryhis waves carry energy,and that the power of thegravitational radiation Ais proportional to thesquare of the third timederivative of thequadrupole moment J ofthe sources

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Gravitational waves: Einstein 1916 - quadrupole formula

Einstein has also shownthat in the linearized theoryhis waves carry energy,and that the power of thegravitational radiation Ais proportional to thesquare of the third timederivative of thequadrupole moment J ofthe sources

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Gravitational waves: Einstein and Rosen 1937 - plane waves are unphysical

Einstein A, Rosen N, Ongravitational waves, Journ. ofFranklin Institute, 223, (1937).

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Gravitational waves: Einstein and Rosen 1937

Rosen’s metric

g = e2φ(dτ2 − dξ2)− u2(e2βdη2 + e−2βdζ2

)with u = τ − ξ, β = β(u), φ = φ(u), φ′ = uβ′2 is a metric

representing empty spacetime Ric(g) = 0 iff

uβ′′ + 2β′ − u2β′3

= 0.Rosen wrongly concluded that this metric can not exist inreality as a spacetime because it contains certain physicalsingularities

He confused coordinate singularity with a true singularity

after suitable coordinate change this can be interpreted as acylindrical wave

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Rosen’s metric




uβ′′ + 2β′ − u2β′3




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Rosen’s metric




uβ′′ + 2β′ − u2β′3




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Rosen’s metric




uβ′′ + 2β′ − u2β′3




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Gravitational waves: main questions

Even after the correct interpretation that this is a cylindrical wavethe questions arrise:

What is a plane gravitational wave in the full theory?

What is a general gravitational wave in the full theory?

Does Rosen’s cylindrical wave carry energy?

Can one have wave solutions of Ric(g) = 0 produced bybounded sources?

These questions had not a satisfactory answer until 1957-1960.

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A naive definition

A naive answer to the plane wave question, would be: agravitational plane wave is a space-time described by ametric, which in some cordinates (t, x , y , z), with t beingtimelike, has metric functions depending on u = t − xonly; preferably these functions to be sin or cos.

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A naive definition


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A naive definition


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A naive definition


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A naive definition


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An (unfair) example

take

g =dt2 − dx2 − dy2 − dz2+

cos(t − x)(2 + cos(t − x))dt2 − 2 cos(t − x)dtdx − 2 cos2(t − x)dtdx + cos2(t − x)dx2

the red terms give a perturbation of the Minkowskimetric, they are oscilatory, riples of the perturbation movewith speed of light

not only the perturbation coefficients satisfy the waveequation, but also Ric(g) = 0 .

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An (unfair) example

take

g =dt2 − dx2 − dy2 − dz2+




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An (unfair) example

take

g =dt2 − dx2 − dy2 − dz2+




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An (unfair) example

take

g =dt2 − dx2 − dy2 − dz2+




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An (unfair) example

take

g =dt2 − dx2 − dy2 − dz2+




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An (unfair) example

take

g =dt2 − dx2 − dy2 − dz2+




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An (unfair) example

take

g =dt2 − dx2 − dy2 − dz2+




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An (unfair) example cntd.

Bad news: the transformation τ = t + sin(t − x) brings themetric from the previous slide to g = dτ2 − dx2 − dy2 − dz2,i.e. the Minkowski metric!

We can produce sinusoidal behaviour of metric coefficients,and movements with speed of light by passing to anappropriate coordinate system!

Conclusion: definition too naive.

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Bondi 1957 - Einstein and Rosen not right

Bondi H, Plane gravitationalwaves in General relativity,Nature, 179, (25.5.1957)

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Bondi, Pirani, Robinson 1958

Bondi H, Pirani F A E,Robinson I Gravitational wavesin General relativity III. Exactplane waves, Proc. R. Soc.London, ser. A, 251,(18.10.1958)

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motivated by the analogy with electromagnetism, whereplane waves have a 5-dimensional group of symmetries theydefined a plane wave in the full GR theory as a solutionto the equations Ric(g) = 0, which has precisely5-dimensional group of symmetriesinspecting Petrov’s list of solutions to Ric(g) = 0 with highsymmetries they found a unique class of solutions that have 5symmetries; the class is given in terms of one free complexfunction f = f (u, ζ), holomorphic in variable ζ, and hasremarkable property which enables to superpose solutionsfrom the classthis enables to produce waves of a sandwich type; they haveshown that a sandwich wave falling on a system of testparticles affects their motion, concluding that plane wavescarry energy.

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Brinkmann 1924

It was a mathematician H. W. Brinkman who first had a generalsolution to Einstein’s equations which now is called plane wave. Itwas in Einstein spaces which are mapped conformally on eachother, Mathem. Annalen, 94, (1925). Totally overlooked by thephysicists!

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Brinkmann 1924


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Brinkmann 1924


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Pirani 1957

Pirani F A E, Invariant Formulation of Gravitational RadiationTheory, Phys. Rev. 105, (18.10.1956)

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Pirani 1957

motivated by Maxwell’s theory, where the radiative solutionsof Maxwell’s equations far from the sources have the curvatureF (A) of the Maxwell’s potential A algebraically special, Piranihad an idea that radiative solutions of the Einstein’sequations far from the sources should have the curvaturetensor Riemann(g) of the metric g algebraically special

Pirani did not know all Petrov types, which were spelled outin full generality by Penrose, much later; he did not made hisstatement precise: it was unclear which Petrov type heattributes to gravitational radiation far from the sources

but the idea that far from the sources garavitationalwave should be of algebraically special Petrov typeturned out to be very important.

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Pirani 1957




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Pirani 1957




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Pirani 1957




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Pirani 1957




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Pirani 1957




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Pirani 1957 - Petrov classification

Nowadays, due to our next hero, we know that far from thesources gravitational wave is of Petrov type N.

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Pirani 1957 - Petrov classification

Nowadays, due to our next hero, we know that far from thesources gravitational wave is of Petrov type N.

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Usefulness of Pirani’s criterion

Quote from Bondi’s Nature paper

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Trautman 1958

[1] Trautman A, Boundary conditions at infinity for physicaltheories Bull. Acad. Polon. Sci., 6, (12.04.1958)[2] Trautman A, Radiation and boundary conditions in thetheory of gravitation, Bull. Acad. Polon. Sci., 6, (12.04.1958).

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Trautman 1958


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Trautman 1958


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Trautman 1958 - radiation is nonlocal

a question from the border between Maxwell’s theory and GR:does a unit charge hang on a thread attached to theceiling of an Einstein’s lift radiates or not? Viewed by theobserver in the lift - NO!, as it is at rest; viewed by anobserver on the Earth - YES!, as it falls down with anacceleration ~g .

this shows that radiation is a nonlocal phenomenon; onecan not apply to it the equivalence principle, since it is alocal law

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Trautman 1958 - gravitational waves via boundary conditions

Trautman’s idea: from all solutions of vacuum Einsteinequations select those that satisfy suitable boundaryconditions at infinity

apropriately reformulate boundary conditions for radiativesolutions of a scalar field known as Sommerfeld’sradiation conditions (Courant, Hilbert, Methods ofMathematical Physics, vol.2, p. 315)

if this is done properly, then such conditions can bestraightforwardly defined in nonlinear theories, in particularin GR.

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Trautman 1958

the first of the two quoted papers is a preparation for themasterpiece, which is the next paper

In the first one Trautman reformulates Sommerfeld’sRADIATION boundary conditions for the scalar relativisticPoisson’s equation to be easily generalized to any filedtheory

as an example he shows how to do it in Maxwell’s theory

The second paper does it for Einstein’s General Relativity

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Trautman 1958





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Trautman 1958





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Trautman 1958





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Trautman 1958





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Trautman 1958: gravitational waves - DEFINITION

In it Trautman defines the boundary conditions for aradiative spacetime in full GR theory. This is a definitionof gravitational radiation. This is in [2], on p. 409,equations (9) and (10).

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Trautman 1958 - reformulation of Einstein’s equations

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Trautman 1958 - energy-momentum of pure gravity

uses von Freud potential 2-form F , to split the Einsteintensor E = Ric(g)− 12Rg into E = dF − 8πt so that theEinstein equations E = 8πT take the form

dF = 8π(T + t) .

Here T is the energy-momentum 3-form.

Since t is a 3-form totally determined by the geometry, heinterprets it as an energy-momentum 3-form of a PUREGRAVITY. This is in [2], on p. 407, equations (1) and (2).

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dF = 8π(T + t) .



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dF = 8π(T + t) .



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dF = 8π(T + t) .



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dF = 8π(T + t) .



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dF = 8π(T + t) .



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Trautman 1958 - 4-momentum of a gravitational system

uses the closed 3-form T + t to define a 4-momentumPµ(σ) of GRAVITATIONAL FIELD attributed to eachspace-like hypersurface σ of a space-time satisfying hisradiative boundary conditions, [2], p. 408, equation (5).

shows thatPµ(σ) is finite andwell defined, i.e. that it does NOT depend on thecoordinate systems adapted to the chosen boundaryconditions, [2], pp. 409-410.

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Trautman 1958: PROOF that gravitational wave CARRY ENERGY

calculates precisely how much of the gravitational energypµ = Pµ(σ1)− Pµ(σ2) contained between the spacelikehypersurfaces σ1 (initial one) and σ2 (final one) escapes toinfinity - or, in nowadays Penrose’s terminology - to scri,[2], p.410-411, equations (16)-(17).shows that p0 is NON-negative, [2], p. 411, remark after(17).

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Trautman 1958: PROOF that grav wave TRAVELS WITH SPEED OF LIGHT

shows that the Ricci tensor of a spacetime satisfying hisradiative conditions, far from the sources, is of the formRicµν = ρkµkν , with k - null vector, [2], p. 411, eq. (20).This in particular means that the gravitational radiation in hisradiative spacetimes travel with speed of light.

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Trautman 1958: PROOF that grav wave IS OF TYPE N

shows that the Riemann tensor of his radiative spacetimes,far from the sources, is of Petrov type N, [2], p. 411, eq.(21). Since far from the sources Riemann = Weyl , this showsthat waves satisfying his boundary conditions satisfy thealgebraic speciality criterion of Pirani.

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Trautman 1958: PROOF that grav wave IS OF TYPE N

shows that the Riemann tensor of his radiative spacetimes,far from the sources, is of Petrov type N, [2], p. 411, eq.(21). Since far from the sources Riemann = Weyl , this showsthat waves satisfying his boundary conditions satisfy thealgebraic speciality criterion of Pirani.

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Robinson, Trautman 1960: gravitational waves FROM BOUNDED SOURCES

[3] Robinson I, Trautman A, Spherical gravitational waves, Phys.Rev. Lett. 4, 431–432 (1960).

Finally, in a common paper with Ivor Robinson, Trautmanfinds EXACT SOLUTIONS of the full system of Einsteinequations satisfying his boundary conditions. Thesolutions describe waves with closed fronts so can beinterpreted as coming from bounded sources.

Robinson-Trautman waves:

g =2r2dζdζP2(u, ζ, ζ)

−2dudr−(4 logP−2r(logP)u−

2m(u)

r

)du2

44(logP) + 12m(logP)u − 4mu = 0, 4 = 2P2∂ζ∂ζ .

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2m(u)

r

)du2


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2m(u)

r

)du2


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Trautman 1958-1960: Importance

Importance of Trautman’s papers [2]-[3]:

first ever precise definition of gravitational radiation inthe full GR theory

first ever general prove that gravitational radiationcarries energy

first explicit examples of metrics describing gravitationalradiation from bounded sources

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Trautman 1958 - King’s College London Lectures

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THANK YOU!

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